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- Start by converting the pdf's to indicator functions
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- Now is defined only when and is defined only when
- Use the convolution formula above to write out the integral
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- Factor out any constants, in this case, a multiplier
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- Factor out the indicator function for into the integral bounds
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- Note that
- Now that have isolated the indicator for z, we can combine the entire integral for that indicator
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- Finally, split the integral into the separate cases based on the remaining indicator function
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- When the integral has no bounds since so the upper bound would be less than which would be .
- When the integral is bound between and since will be at least but less than
- As you can see there is a pattern here, it goes as follows:
- Given you will have